Perturbative Constraints Research Articles

Use of polynomial chaos expansions in the design of uncertainty-robust matched field processing algorithms

The polynomial chaos (PC) expansion method has been recently applied to estimating the statistical properties of underwater acoustic propagation in the presence of environmental uncertainty. Here we use PC estimates of the field covariance structure to design uncertainty robust match field processing (MFP) weights using Krolik’s minimum variance beamformer with sound-speed perturbation constraints (MV-SPC) [J. Acoust. Soc. Am. 92 (3), 1408–1419 (1992)]. The idea behind the MV-SPC beamformer is to realize much of the high sidelobe rejection of MV processors in environments with environmental variability by opening up the signal model to include uncertainty effects. Here we compare the performance of MV-SPC designed with an adiabatic signal uncertainty model to the same beamformer designed with the PC signal model for realizations of the signal vector obtained with fully coupled propagation models, with results showing the superiority of the PC approach. [Work supported by ONR.]

Electromagnetic Target Detection in Uncertain Media: Time-Reversal and Minimum-Variance Algorithms

An experimental study is performed on imaging targets that are situated in a highly scattering environment, employing electromagnetic time-reversal methods. A particular focus is placed on performance when the electrical properties of the background environment (medium) are uncertain. It is assumed that the (unknown) medium characteristic of the scattered fields represents one sample from an underlying random process, with this random process representing our uncertainty in the media properties associated with the scattering measurement. While the specific Green's function associated with the scattered fields is unknown, we assume access to an ensemble of Green's functions sampled from the aforementioned distribution. This ensemble of Green's functions may be used in several ways to mitigate uncertainty in the true Green's function. Specifically, when performing time-reversal imaging, we consider a Green's function as a representative of the average of the ensemble, as well as Green's functions based on a principal components analysis of the ensemble. We also develop a wideband minimum-variance beamformer with environment perturbation constraints, in which the unknown Green's function is constrained to reside in a subspace spanned by the Green's function ensemble. These algorithms are examined using electromagnetic scattering data measured in a canonical set of laboratory experiments. The qualitative performance of the different techniques is presented in the form of images, with quantitative results presented in the form of receiver operating characteristic performance

Robust matched-field processor in the presence of geoacoustic inversion uncertainty

This presentation examines the performance of a matched-field processor incorporating geoacoustic inversion uncertainty. Uncertainty of geoacoustic parameters is described via a joint posterior probability distribution (PPD) of the estimated environmental parameters, which is found by formulating and solving the geoacoustic inversion problem in a Bayesian framework. The geoacoustic inversion uncertainty is mapped into uncertainty in the acoustic pressure field. The resulting acoustic field uncertainty is incorporated in the matched-field processor using the minimum variance beamformer with environmental perturbation constraints (MV-EPC). The constraints are estimated using the ensemble of acoustic pressure fields derived from the PPD of the estimated environmental parameters. Using a data set from the ASIAEX 2001 East China Sea experiment, tracking performance of the MV-EPC beamformer is compared with the Bartlett beamformer using the best-fit model.

Constructing the fermion-boson vertex in three-dimensional QED

We derive perturbative constraints on the transverse part of the fermion-boson vertex in massive QED3 through its one loop evaluation in an arbitrary covariant gauge. Written in a particular form, these constraints naturally lead us to the first non-perturbative construction of the vertex, which is in complete agreement with its one loop expansion in all momentum regimes. Without affecting its one-loop perturbative properties, we also construct an effective vertex in such a way that the unknown functions defining it have no dependence on the angle between the incoming and outgoing fermion momenta. Such a vertex should be useful for the numerical study of dynamical chiral symmetry breaking, leading to more reliable results.

RADION POTENTIAL AND BRANE DYNAMICS

We examine the cosmology of Randall–Sundrum model in a dynamic setting where scalar fields are present in the bulk as well as the branes. This generates a mechanism similar to that of Goldberger–Wise for radion stabilization and the recovery of late-time cosmology features on the branes. Due to the induced radion dynamics, the inflating branes roll towards the minimum of the radion potential, thereby exiting inflation and reheating the universe. In the slow roll part of the potential, the TeV branes have maximum inflation rate and energy as their coupling to the radion and bulk modes have minimum suppression. Hence, when rolling down the steep end of the potential towards the stable point, the radion field (which appears as the inflaton of the effective 4-D theory in the branes) decays very fast and reheats the universe. This process results in a decrease of the brane's canonical vacuum energy, Λ4. However, at the minimum of the potential Λ4 is small but not necessarily zero and the fine-tuning issue remains. Density perturbation constraints introduce an upper bound on Λ4. Due to the large radion mass and strong suppression to the bulk modes, moduli problems and bulk reheating do not occur. The reheat temperature and a sufficient number of e-folding constraints for the brane-universe are also satisfied. The model therefore recovers the radiation dominated FRW universe.

Comparison of robust matched-field processing algorithms in a real, shallow water environment

In recent years, a number of matched-field processing (MFP) algorithms have been proposed to mitigate the ever-present problem of mismatch. This paper compares the source localization performance of six processors—Bartlett, Midpoint, Minimum Variance (MV) Distortionless Response, White Noise Constrained, MV with Neighboring Location Constraints, and MV with Environmental Perturbation Constraints—in a real, shallow water environment. MFP was performed on an 84.4-m aperture, 16-element subset of a MPL vertical line array deployed from the R/P FLIP west of Point Loma in approximately 200-m water. A continuous wave source, emitting ten tonals from 53–197 Hz, was towed over both range-independent and range-dependent tracks at source ranges of 1–7 km and at source depths of 60–130 m. First, MFP results utilizing a detailed geoacoustic model of the region, water column sound speed profiles measured during the experiment, and a priori knowledge of array tilt, demonstrate general processor characteristics. Second, MFP results which assume: (1) reasonable though further mismatched geoacoustic parameters; (2) water column sound speed profiles based upon historical data; and (3) no a priori knowledge of array tilt, illustrate processor performance under various mismatch conditions likely to be encountered in realistic applications.

Variational principle and a perturbative solution of non-linear string equations in curved space

String dynamics in a curved space-time is studied on the basis of an action functional including the small parameter of rescaled tension ε = γ/ α′, where γ is a metric parametrizing constant. A rescaled slow world-sheet time T = ετ is introduced, and general covariant non-linear string equations are derived. It is shown that in the first order of an ε-expansion these equations are reduced to the known equation for geodesic deviation but complemented by a string oscillatory term. These equations are solved for the de Sitter and Friedmann-Robertson-Walker spaces. The primary string constraints are found to be split into a chain of perturbative constraints and their conservation and consistency are proved. It is established that in the proposed realization of the perturbative approach the string dynamics in the de Sitter space is stable for a large Hubble constant H( α′ H 2 ⪢ 1).

SUSY SU(5) and hybrid inflation

We consider a scenario of hybrid inflation in the framework of the minimal supersymmetric SU(5) model, compatible, for a quite wide range of initial conditions for inflation scalar fields, with inflation efficiency, monopole dilution and perturbation constraints.

Reduction in computational requirements with matched field processing

The sensitivity of matched field techniques to source location is a significant benefit from a localization accuracy perspective. Unfortunately, the implementation of matched field techniques in real-time systems is hampered by the computationally demanding hypothesis testing of source location in range and depth. Furthermore, operational requirements of such systems generally do not demand a high degree of accuracy. The Minimum Variance with Environmental Perturbation Constraints (MV-EPC) processor developed by Krolik [J. Acoust. Soc. Am. 92, 1408–1419 (1992)] has been used to reduce the sensitivity to environmental uncertainty. This same approach can be used to reduce the sensitivity to source location, thereby reducing the number of source location hypotheses. This produces a computational tractable approach to real-time, robust matched field processing. Illustrative simulations and real data results using SWellEX-96 data are presented.

Relativistic conservation laws and integral constraints for large cosmological perturbations

For every mapping of a perturbed spacetime onto a background and with any vector field $\xi$ we construct a conserved covariant vector density $I(\xi)$, which is the divergence of a covariant antisymmetric tensor density, a "superpotential". $ I(\xi)$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I(\xi)$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating conserved vectors over a part $\Si$ of a hypersurface $S$ of the background, which spans a two-surface $\di\Si$, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on $\Si$ and, on the other hand, the boundary values on $\di\Si$. We show that there are as many such integral relations as there are different mappings, $\xi$'s, $\Si$'s and $\di\Si$'s. For given boundary values on $\di\Si$, the integral relations may be interpreted as integral constraints (e.g., those of Traschen) on local initial data including the energy-momentum perturbations. Conservation laws expressed in terms of Killing fields $\Bar\xi$ of the background become "physical" conservation laws. In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. ...

Matched-field source tracking with SWELLEX-1 data

The optimal approach to tracking a moving source in an uncertain environment is one which incorporates a priori knowledge about both the continuity of the propagation environment and the nature of the source movement. The optimum uncertain field tracking algorithm (OUFTA) [S. L. Tantum and L. W. Nolte, ‘‘Tracking and localizing a moving source in an uncertain shallow water environment,’’ J. Acoust. Soc. Am. (submitted)] extends the optimum uncertain field processor (OUFP) [A. M. Richardson and L. W. Nolte, J. Acoust. Soc. Am. 89, 2280–2284 (1991)] to include modeling of the source motion as a Markov process. This differs from the suboptimal approach which performs a series of independent source localizations and then combines the results to estimate the path taken by the source. The performance of the OUFTA and the suboptimal approach is evaluated using the SWELLEX-1 data collected off the San Diego coast. Results using the SWELLEX-1 data are also presented which show how this optimal philosophy can be used to guide the modification of some popular beamformers, such as the minimum variance adaptive beamformer with environmental perturbation constraints (MV-EPC) [J. L. Krolik, J. Acoust. Soc. Am. 92, 1408–1419 (1992)], to incorporate source motion. [Work supported by ONR.]

Matched-field source localization in a random ocean

This paper is devoted to the statistical approach to matched-field processing (MFP). In Sec. I, a statistical model of the Green’s function of a random oceanic waveguide is described using the mean field and the second moment. In Sec. 2, using Bayesian approach, optimum target detection and localization algorithms matched to a deterministic and a stochastic ocean are derived. First, the optimum decisive statistic (log-likelihood ratio) is derived for a deterministic ocean. In the general case, ocean parameter fluctuations make the Green’s function random and so we need to average the log-likelihood ratio. The MFP algorithms for a random ocean waveguide developed on the basis of the Bayesian approach are less precise than the algorithms for a deterministic oceanic waveguide model, but they are more robust. In Sec. 3 a comparative analysis of the proposed MFP methods and the known suboptimum MFP techniques is carried out. The ambiguity surfaces for the proposed algorithms, Bartlett, Minimum Variance (MV), MV with neighboring location constraints, and MV with sound-speed perturbation constraints methods are presented. The proposed MFP method outperformed other algorithms under random ocean conditions, giving the least bias error and the lowest sidelobes.

Robust wideband matched-field processing with a short vertical array

This paper addresses the problem of matched-field passive localization of broadband underwater acoustic sources with a short vertical array. In previous work, incoherent averaging of narrow-band Bartlett ambiguity surfaces has been proposed to improve the robustness of matched-field processing (MFP) to environmental uncertainties. While computationally efficient, the effectiveness of this approach is dependent on the type of environmental mismatch present. In this paper, to provide robust source localization performance for short arrays, an alternative wideband MFP method is proposed which utilizes the fourth-order statistics of the data. The method is “coherent” in that it exploits cross-frequency dependencies in the random acoustic channel response. The proposed coherent wideband minimum variance method with environmental perturbation constraints (MV-EPC) consists of minimizing the squared output power of a beamformer subject to constraints defined across the signal band. The constraints are designed to provide robustness over an ensemble of random environmental realizations. Monte Carlo simulation results using a canonical shallow-water scenario indicate that the coherent wideband MV-EPC method yields improved probability of correct localization performance versus incoherent averaging of narrow-band Bartlett ambiguity functions with a short array above a threshold signal-to-noise ratio.

Error-based constraints for efficient learning of pattern recognition problems

The depth of a feedforward network is related to its ability to solve complex pattern recognition problems. An error-based layer dependent weight perturbation constraint is proposed for efficiently training deep networks using the error backpropagation algorithm. The performance of this scheme is compared with the generic backpropagation as well as the delta-bar-delta rule for training feedforward networks.

Minimal four-family supergravity model.

We investigate the phenomenology of minimal four-family MSSM supergravity theories containing an additional generation of heavy fermions along with their superpartners. We demand: gauge coupling constant unification at high energy scales; perturbative values for all Yukawa couplings for energy scales $\leq \mgut$; radiative electroweak symmetry breaking via renormalization group evolution; and a neutral LSP. The perturbative constraints imply a light fourth-family quark and lepton spectrum, and $\tanb\lsim 3$. The lightest CP-even Higgs mass is increased. Fourth-family Yukawa coupling contributions to the evolution of scalar masses lead to unexpected mass hierarchies; \eg\ the $\staupone$ is generally the lightest slepton and the lightest squark is the $\wt\bpr_1$. A significant lower bound is placed on the gluino mass by the requirement that the $\staupone$ not be the LSP. Sleptons of the first two families are much more massive compared to the LSP and other neutralinos and charginos than in the three-family models, and could easily lie beyond the reach of a $\sqrt s=500\gev$ $\epem$ collider. Relations between slepton masses and gaugino masses are shown to be very sensitive to the presence of a fourth generation. The most important near-future experimental probes of the four-family models are reviewed. A scenario with $\mt\sim\mw$ and $t\rta {\wt t_1}\cnone$ is shown to be inconsistent with universal soft-SUSY-breaking boundary conditions. Full four-family evolution of $\alpha_s$ is shown to lead to a significant enhancement in inclusive jet and di-jet spectra at Tevatron energies when all sparticle masses are near their lower bounds.

The performance of matched-field beamformers with Mediterranean vertical array data

Minimum variance (MV) adaptive beamforming has been widely proposed for matched-field processing because it provides a means of suppressing ambiguous beampattern sidelobes. A difficulty with MV methods, however, is their sensitivity to signal wavefront mismatch. In this work, the performance of three robust MV methods and the Bartlett beamformer is evaluated using vertical array data from the Mediterranean Sea collected by the NATO SACLANT Centre. The three MV methods considered are: (1) the reduced MV beamformer (RMV), (2) the MV beamformer with neighborhood location constraints (MV-NLC), (3) the MV beamformer with environmental perturbation constraints (MV-EPC). While the Bartlett, RMV, and MV-NLC methods assume the ocean environment is known precisely, the MV-EPC method models the environment as being random with known statistics. Experimental and companion simulation results indicate that for modest environmental uncertainty, the MV-EPC beamformer achieves a higher probability of correct localization and better sidelobe performance than the other three methods.

REDUCED PHASE SPACE QUANTIZATION OF ASHTEKAR’S GRAVITY ON A DE SITTER BACKGROUND

We solve perturbative constraints and eliminate gauge freedom for Ashtekar’s gravity on a de Sitter background. We show that the reduced phase space consists of transverse, traceless, symmetric fluctuations of the triad and transverse, traceless, symmetric fluctuations of the connection. A part of the gauge freedom corresponding to the conformal Killing vectors of the three-manifold can be fixed only by imposing conditions on the Lagrange multiplier. The reduced phase space is equivalent to that of ADM gravity on the same background.

Minimum variance matched‐field source localization with SACLANT Mediterranean sea trial data

The minimum variance (MV) adaptive beamformer has been widely proposed for matched‐field processing because it provides a means of suppressing ambiguous beampattern ‘‘sidelobes.’’ In this paper, the performance of three robust MV methods is evaluated using vertical array data from the Mediterranean Sea collected by the NATO SACLANT Centre north of Elba [D. F. Gingras and P. Gerstoft, 3234 (1994)]. The three MV methods considered are (1) the reduced MV method (RMV) [C. L. Byrne et al., 2493–2502 (1990)], (2) the MV method with neighborhood location constraints (MV‐NLC) [H. Schmidt et al., 1851–62 (1990)], and (3) the MV method with environmental perturbation constraints (MV‐EPC) [J. Krolik, 1408–19 (1992)]. Real data results indicate that each approach has different merits. Projecting the data onto the reduced space defined by the modal eigenfunctions as in the RMV is necessary to avoid instabilities caused by highly correlated noise. Using neighboring location constraints as in the MV‐NLC reduces the need to finely sample the ambiguity surface. And using constraints derived from the second‐order statistics of the signal wavefront averaged over an ensemble of perturbed environmental parameters as in the MV‐EPC increases robustness to environmental mismatch. [Work supported by ONR.]

COMPUTATIONALLY EFFICIENT MONTE CARLO ESTIMATION ALGORITHMS FOR MATCHED FIELD PROCESSING IN UNCERTAIN OCEAN ENVIRONMENTS

In this paper, Monte Carlo estimation techniques are presented for computationally efficient implementation of two methods for matched field source localization in uncertain ocean channels. In the Optimal Uncertain Field Processor (OUFP), Monte Carlo integration is used to integrate out the environmental parameters and thus estimate the a posteriori distribution of the source location parameters. In the Minimum Variance Beamformer with Environmental Perturbation Constraints (MV-EPC), Monte Carlo estimation of the signal correlation matrix averaged over the ensemble of environmental realizations is used to estimate the beamformer constraints. Using the OUFP, detection performance bounds are evaluated which indicate that source position uncertainty affects performance much more than environmental uncertainty. An upper bound on source localization performance is also obtained indicating that for short observation times a threshold signal-to-noise ratio (SNR) exists, dependent upon environmental uncertainty, below which source localization performance rapidly degrades. Among robust minimum variance beamforming methods, the MV-EPC method demonstrated superior probability of correct localization (PCL), both in single source scenarios and in the presence of interference. The OUFP at high SNR and the MV-EPC at large observation times both achieved near perfect source localization performance, although for large environmental uncertainty the OUFP provides an upper bound on P CL .

Environmental data assimilation in matched-field processing with a random normal mode model

Numerical acoustic propagation models used in matched-field processing typically assume that ocean environmental parameters, such as the range-dependent sound-speed profile, are either known exactly or are characterized by stationary prior statistics. However, neither of these modeling approaches is well suited for accurately assimilating measurements of the correct ocean realization into the calculation of point-source acoustic wave front statistics. In this paper, environmental data is assimilated into a random adiabatic normal mode propagation model by calculating the conditional correlation matrix of an acoustic wave front given a set of contemporaneous sound-speed profile measurements made at different points in space. First-order perturbation theory is used to map the conditional statistics of Gaussian-distributed excess sound-speed variations into an estimate of the conditional point-source wave front correlation matrix. Minimum variance beamforming with environmental perturbation constraints [J. L. Krolik, J. Acoust. Soc. Am. 92, 1408–1419 (1992)] conditioned on current environmental measurements is demonstrated as a means of exploiting this random wave front model to achieve improved matched-field source localization performance. [Work supported by ONR.]